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In model theory, interpretation of a structure ''M'' in another structure ''N'' (typically of a different signature) is a technical notion that approximates the idea of representing ''M'' inside ''N''. For example every reduct or definitional expansion of a structure ''N'' has an interpretation in ''N''. Many model-theoretic properties are preserved under interpretability. For example if the theory of ''N'' is stable and ''M'' is interpretable in ''N'', then the theory of ''M'' is also stable. ==Definition== An interpretation of ''M'' in ''N'' with parameters (or without parameters, respectively) is a pair where ''n'' is a natural number and is a surjective map from a subset of ''Nn'' onto ''M'' such that the -preimage (more precisely the -preimage) of every set ''X'' ⊆ ''Mk'' definable in ''M'' by a first-order formula without parameters is definable (in ''N'') by a first-order formula with parameters (or without parameters, respectively). Since the value of ''n'' for an interpretation is often clear from context, the map itself is also called an interpretation. To verify that the preimage of every definable (without parameters) set in ''M'' is definable in ''N'' (with or without parameters), it is sufficient to check the preimages of the following definable sets: * the domain of ''M''; * the diagonal of ''M''; * every relation in the signature of ''M''; * the graph of every function in the signature of ''M''. In model theory the term ''definable'' often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term ''0-definable''. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a 0-interpretation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Interpretation (model theory)」の詳細全文を読む スポンサード リンク
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